arXiv subjects

Christian Coester

Publications and source records attributed to Christian Coester.

At least 19 recordsLinked to original sources

arXiv2026

Competitive Online Clustering with Movement Cost

We study the problem of online clustering with movement cost, which generalizes the fundamental $k$-server problem. At each time step, the algorithm is given a set of points in a metric space and must choose $k$ cluster centers. The cost incurred is the sum of the clustering cost at each time step and the total distance that the centers move over time. Prior work established competitive algorithms for the case $k\in\{1,2\}$ on the line metric with the $k$-median clustering objective; under the $k$-median objective, this problem is also known as facility reallocation. We substantially generalize these results to arbitrary $k$, general metric spaces, and a broad class of clustering objectives. Specifically, we consider clustering costs defined as a norm of the distance vector from points to their nearest center, which subsumes the $k$-median, $k$-center, and (square root of) $k$-means objectives, and provide an asymptotically optimal $O(k)$-competitive randomized algorithm against adaptive online adversaries. The algorithm is based on a reduction to the $k$-server problem, establishing a tight connection between the problems. In this reduction, $k$-server requests are generated by Poisson processes, whose arrival rates evolve in continuous time and are driven by a dual representation of the clustering costs. Our method of using Poissonization in this context may also be of independent interest. We demonstrate another use of the idea by deriving similar reductions for batched set chasing problems. We also consider a different family of clustering objectives defined as a norm of the vector of cluster radii, which subsumes the sum-$k$-radii and the $k$-center objectives, and give a tight $O(k)$-competitive deterministic algorithm on the line.

cs.DS↗

arXiv2026

The $k$-server conjecture is true

The $k$-server conjecture states that a deterministic online algorithm can achieve competitive ratio $k$ on every metric space. We prove the conjecture. Specifically, we show that the work function algorithm satisfies it. Our proof uses a natural algebraic representation of the work function as a matrix, which encodes all feasible paths to reach a configuration. In this representation, the minimum and addition operations arising in the definition of optimal costs correspond to addition and multiplication of formal expressions, and each work function value corresponds to the determinant of $k$ columns of the matrix. A request arrival updates the representation via a change of basis and row replacement. The amortized analysis is based on a potential function defined in terms of a larger matrix whose coordinates are pairs of coordinates of the original matrix representation.

cs.DS↗

arXiv2026

Primal-Dual Online Algorithms for the Parking Permit Problem

The Parking Permit Problem (PPP), first studied by Meyerson, is a classic online problem generalizing the ski rental problem. We re-examine the PPP using the primal-dual scheme, obtaining simple algorithms with superior performance guarantees. Unlike previous work, which relied on reductions that degraded competitive ratios, we work with the problem's structure directly. We also provide near-matching lower bounds. Using the primal-dual framework, we find the PPP's deterministic competitive ratio exactly, and the randomized competitive ratio within an additive constant.

cs.DS↗

arXiv2026

Learning-Augmented Online Minimization with Dual Predictions

We present learning-augmented algorithms for two general classes of online minimization problems: metrical task systems and laminar set cover. Both algorithms achieve improved theoretical guarantees using machine-learned predictions of an optimal solution to the dual linear program. Unlike optimal primal solutions, which can change drastically under tiny instance perturbations, these dual solutions are much more stable, which ensures the existence of good (and learnable) predictions for families of similar instances. While previous work has used dual predictions in offline settings and for online maximization problems, our algorithms are, to the best of our knowledge, the first demonstration that such dual predictions can be effective for online minimization. Our theoretical results are complemented by experiments on the $k$-server problem and the parking permit problem.

cs.DS↗

arXiv2026

Online Monotone Metric Embeddings

Metric embeddings into structured spaces, particularly hierarchically well-separated trees (HSTs), are a fundamental tool in the design of online algorithms. In the classical online embedding setting, points arrive sequentially and must be embedded irrevocably upon arrival, resulting in strong distortion lower bounds of $Ω(\min(n, \log n\log Δ))$, where $n$ is the number of points and $Δ$ their aspect ratio. We propose a novel relaxation, online monotone metric embeddings, which allows distances between embedded points in the target space to decrease monotonically over time. Such relaxed embeddings remain compatible with many online algorithms. Moreover, this relaxation breaks existing lower bound barriers, enabling embeddings into HSTs with distortion $O(\log^2 n)$. We also study a dynamic variant, where points may both arrive and depart, seeking distortion guarantees in terms of the maximum number $l$ of simultaneously present points. For traditional embeddings, such bounds are impossible, and this limitation persists even for deterministic monotone embeddings. Surprisingly, probabilistic monotone embeddings allow for $O(l \log l)$ distortion, which is nearly optimal given an $Ω(l)$ lower bound.

cs.DS↗

arXiv2026

Chasing Small Sets Optimally Against Adaptive Adversaries

We study deterministic online algorithms for the problem of chasing sets of cardinality at most $k$ in a metric space, also known as metrical service systems and equivalent to width-$k$ layered graph traversal. We resolve the 30-year-old gap of $Ω(2^k)\cap O(k2^k)$ on the competitive ratio of this problem by giving an $O(2^k)$-competitive deterministic algorithm. This bound is optimal even among randomized algorithms against adaptive adversaries. We also (slightly) improve the deterministic lower bound to $D_k$, defined recursively by $D_1=1$ and $D_{k+1}=2D_k+\sqrt{8+8D_k}+3$, which we conjecture to be exactly tight. For $k=3$, we provide a matching upper bound of $D_3$. Our results imply slightly improved upper and lower bounds for distributed asynchronous collective tree exploration and for the $k$-taxi problem, respectively. Our algorithm generalizes the classical doubling strategy, previously known to be optimal for $k=2$. The previous best bound for general $k$ was achieved by the generalized work function algorithm (WFA), and was known to be tight for WFA. Our improved bound therefore implies that WFA is sub-optimal for chasing small sets.

cs.DS↗

arXiv2026

Randomized $k$-server in polynomial time

We study the design of computationally efficient randomized algorithms for the $k$-server problem. Existing randomized algorithms with the best known competitive ratios are, on the one hand, inherently implicit and, on the other hand, employ a rounding scheme that maintains a distribution over exponentially many configurations. In this work, we introduce a derandomization framework that transforms any randomized $k$-server algorithm on a hierarchically separated tree into one that uses only $O(\log k)$ random bits for request sequences of arbitrary length; hence maintaining a distribution over only polynomially many server configurations. Leveraging this black-box derandomization, we obtain the first polynomial-time randomized $k$-server algorithm on arbitrary $n$-point metrics with a polylogarithmic competitive ratio. Our results also have implications for the advice complexity of the $k$-server problem.

cs.DS↗

arXiv2026

Transposition is Nearly Optimal for IID List Update

The list update problem is one of the oldest and simplest problems in online algorithms: A set of items must be maintained in a list while requests to these items arrive over time. Whenever an item is requested, the algorithm pays a cost equal to the position of the item in the list. In the i.i.d. model, where requests are drawn independently from a fixed distribution, the static ordering by decreasing access probabilities $p_1\ge p_2\ge \dots \ge p_n$ achieves the minimal expected access cost OPT$=\sum_{i=1}^n ip_i$. However, $p$ is typically unknown, and approximating it by tracking access frequencies creates undesirable overheads. We prove that the Transposition rule (swap the requested item with its predecessor) has expected access cost at most OPT$+1$ in its stationary distribution. This confirms a 50-year-old conjecture by Rivest up to an unavoidable additive constant. More abstractly, it yields a purely memoryless procedure to approximately sort probabilities via sampling. Our proof is based on a decomposition of excess cost, and its technical core is a "sign-eliminating" combinatorial injection to witness nonnegativity of a constrained multivariate polynomial.

cs.DS↗

arXiv2025

Early science acceleration experiments with GPT-5

AI models like GPT-5 are an increasingly valuable tool for scientists, but many remain unaware of the capabilities of frontier AI. We present a collection of short case studies in which GPT-5 produced new, concrete steps in ongoing research across mathematics, physics, astronomy, computer science, biology, and materials science. In these examples, the authors highlight how AI accelerated their work, and where it fell short; where expert time was saved, and where human input was still key. We document the interactions of the human authors with GPT-5, as guiding examples of fruitful collaboration with AI. Of note, this paper includes four new results in mathematics (carefully verified by the human authors), underscoring how GPT-5 can help human mathematicians settle previously unsolved problems. These contributions are modest in scope but profound in implication, given the rate at which frontier AI is progressing.

cs.CL↗

arXiv2025

Online 3-Taxi on General Metrics

The online $k$-taxi problem, introduced in 1990 by Fiat, Rabani and Ravid, is a generalization of the $k$-server problem where $k$ taxis must serve a sequence of requests in a metric space. Each request is a pair of two points, representing the pick-up and drop-off location of a passenger. In the interesting ''hard'' version of the problem, the cost is the total distance that the taxis travel without a passenger. The problem is known to be substantially harder than the $k$-server problem, and prior to this work even for $k=3$ taxis it has been unknown whether a finite competitive ratio is achievable on general metric spaces. We present an $O(1)$-competitive algorithm for the $3$-taxi problem.

cs.DS↗

arXiv2025

Smoothed Analysis of Online Metric Problems

We study three classical online problems -- $k$-server, $k$-taxi, and chasing size $k$ sets -- through a lens of smoothed analysis. Our setting allows request locations to be adversarial up to small perturbations, interpolating between worst-case and average-case models. Specifically, we show that if the metric space is contained in a ball in any normed space and requests are drawn from distributions whose density functions are upper bounded by $1/σ$ times the uniform density over the ball, then all three problems admit polylog$(k/σ)$-competitive algorithms. Our approach is simple: it reduces smoothed instances to fully adversarial instances on finite metrics and leverages existing algorithms in a black-box manner. We also provide a lower bound showing that no algorithm can achieve a competitive ratio sub-polylogarithmic in $k/σ$, matching our upper bounds up to the exponent of the polylogarithm. In contrast, the best known competitive ratios for these problems in the fully adversarial setting are $2k-1$, $\infty$ and $Θ(k^2)$, respectively.

cs.DS↗

arXiv2024

Shortest Paths without a Map, but with an Entropic Regularizer

In a 1989 paper titled "shortest paths without a map", Papadimitriou and Yannakakis introduced an online model of searching in a weighted layered graph for a target node, while attempting to minimize the total length of the path traversed by the searcher. This problem, later called layered graph traversal, is parametrized by the maximum cardinality $k$ of a layer of the input graph. It is an online setting for dynamic programming, and it is known to be a rather general and fundamental model of online computing, which includes as special cases other acclaimed models. The deterministic competitive ratio for this problem was soon discovered to be exponential in $k$, and it is now nearly resolved: it lies between $Ω(2^k)$ and $O(k2^k)$. Regarding the randomized competitive ratio, in 1993 Ramesh proved, surprisingly, that this ratio has to be at least $Ω(k^2 / \log^{1+ε} k)$ (for any constant $ε> 0$). In the same paper, Ramesh also gave an $O(k^{13})$-competitive randomized online algorithm. Between 1993 and the results obtained in this paper, no progress has been reported on the randomized competitive ratio of layered graph traversal. In this work we show how to apply the mirror descent framework on a carefully selected evolving metric space, and obtain an $O(k^2)$-competitive randomized online algorithm. This matches asymptotically an improvement of the aforementioned lower bound (Bubeck, Coester, Rabani; STOC 2023), which we announced (among other results) after the initial publication of the results here.

cs.DS↗

arXiv2024

Learning-Augmented Priority Queues

Priority queues are one of the most fundamental and widely used data structures in computer science. Their primary objective is to efficiently support the insertion of new elements with assigned priorities and the extraction of the highest priority element. In this study, we investigate the design of priority queues within the learning-augmented framework, where algorithms use potentially inaccurate predictions to enhance their worst-case performance. We examine three prediction models spanning different use cases, and show how the predictions can be leveraged to enhance the performance of priority queue operations. Moreover, we demonstrate the optimality of our solution and discuss some possible applications.

cs.DS↗

arXiv2024

Unweighted Layered Graph Traversal: Passing a Crown via Entropy Maximization

Introduced by Papadimitriou and Yannakakis in 1989, layered graph traversal is a central problem in online algorithms and mobile computing that has been studied for several decades, and which now is essentially resolved in its original formulation. In this paper, we demonstrate that what appears to be an innocuous modification of the problem actually leads to a drastic (exponential) reduction of the competitive ratio. Specifically, we present an algorithm that is $O(\log^2 w)$-competitive for traversing unweighted layered graphs of width $w$. Our algorithm chooses the agent's position simply according to the probability distribution over the current layer that maximizes the sum of entropies of the induced distributions in the preceding layers.

cs.DS↗

arXiv2023

Mixing predictions for online metric algorithms

A major technique in learning-augmented online algorithms is combining multiple algorithms or predictors. Since the performance of each predictor may vary over time, it is desirable to use not the single best predictor as a benchmark, but rather a dynamic combination which follows different predictors at different times. We design algorithms that combine predictions and are competitive against such dynamic combinations for a wide class of online problems, namely, metrical task systems. Against the best (in hindsight) unconstrained combination of $\ell$ predictors, we obtain a competitive ratio of $O(\ell^2)$, and show that this is best possible. However, for a benchmark with slightly constrained number of switches between different predictors, we can get a $(1+ε)$-competitive algorithm. Moreover, our algorithms can be adapted to access predictors in a bandit-like fashion, querying only one predictor at a time. An unexpected implication of one of our lower bounds is a new structural insight about covering formulations for the $k$-server problem.

cs.LG↗

arXiv2023

Sorting with Predictions

We explore the fundamental problem of sorting through the lens of learning-augmented algorithms, where algorithms can leverage possibly erroneous predictions to improve their efficiency. We consider two different settings: In the first setting, each item is provided a prediction of its position in the sorted list. In the second setting, we assume there is a "quick-and-dirty" way of comparing items, in addition to slow-and-exact comparisons. For both settings, we design new and simple algorithms using only $O(\sum_i \log η_i)$ exact comparisons, where $η_i$ is a suitably defined prediction error for the $i$th element. In particular, as the quality of predictions deteriorates, the number of comparisons degrades smoothly from $O(n)$ to $O(n\log n)$. We prove that the comparison complexity is theoretically optimal with respect to the examined error measures. An experimental evaluation against existing adaptive and non-adaptive sorting algorithms demonstrates the potential of applying learning-augmented algorithms in sorting tasks.

cs.DS↗

arXiv2023

The Randomized $k$-Server Conjecture is False!

We prove a few new lower bounds on the randomized competitive ratio for the $k$-server problem and other related problems, resolving some long-standing conjectures. In particular, for metrical task systems (MTS) we asympotically settle the competitive ratio and obtain the first improvement to an existential lower bound since the introduction of the model 35 years ago (in 1987). More concretely, we show: 1. There exist $(k+1)$-point metric spaces in which the randomized competitive ratio for the $k$-server problem is $Ω(\log^2 k)$. This refutes the folklore conjecture (which is known to hold in some families of metrics) that in all metric spaces with at least $k+1$ points, the competitive ratio is $Θ(\log k)$. 2. Consequently, there exist $n$-point metric spaces in which the randomized competitive ratio for MTS is $Ω(\log^2 n)$. This matches the upper bound that holds for all metrics. The previously best existential lower bound was $Ω(\log n)$ (which was known to be tight for some families of metrics). 3. For all $k<n\in\mathbb N$, for *all* $n$-point metric spaces the randomized $k$-server competitive ratio is at least $Ω(\log k)$, and consequently the randomized MTS competitive ratio is at least $Ω(\log n)$. These universal lower bounds are asymptotically tight. The previous bounds were $Ω(\log k/\log\log k)$ and $Ω(\log n/\log \log n)$, respectively. 4. The randomized competitive ratio for the $w$-set metrical service systems problem, and its equivalent width-$w$ layered graph traversal problem, is $Ω(w^2)$. This slightly improves the previous lower bound and matches the recently discovered upper bound. 5. Our results imply improved lower bounds for other problems like $k$-taxi, distributed paging and metric allocation. These lower bounds share a common thread, and other than the third bound, also a common construction.

cs.DS↗

arXiv2023

Online metric algorithms with untrusted predictions

Machine-learned predictors, although achieving very good results for inputs resembling training data, cannot possibly provide perfect predictions in all situations. Still, decision-making systems that are based on such predictors need not only to benefit from good predictions but also to achieve a decent performance when the predictions are inadequate. In this paper, we propose a prediction setup for arbitrary metrical task systems (MTS) (e.g., caching, k-server and convex body chasing) and online matching on the line. We utilize results from the theory of online algorithms to show how to make the setup robust. Specifically for caching, we present an algorithm whose performance, as a function of the prediction error, is exponentially better than what is achievable for general MTS. Finally, we present an empirical evaluation of our methods on real world datasets, which suggests practicality.

cs.DS↗